� Hopf Bifurcations in 2D

I 2 ways for stable fixed point to lose stability:

• 1 real eigenvalue passes through λ = 0(zero-eigenvalue bifurcations),

• 2 complex conjugate eigenvalues cross into right half plane(Hopf bifurcations).

I Supercritical Hopf bifurcation occursif exponentially damped oscillation changes to growth at µc, andbecomes small limit cycle oscillation about formerly steady state.

� Supercritical Hopf Bifurcations in 2D

I Supercritical Hopf bifurcation occurs when stable spiral changesinto unstable spiral surrounded by small limit cycle.

I For example:

r = µr − r3, θ = ω + br2,

where µ controls stability at origin,ω gives frequency of infinitismal oscillations,b determines dependence of frequency on larger amplitude.

I Eigenvalues λ = µ± iω cross imaginary axis from left to right.

� Rules of Thumb for Supercritical Hopf Bifurcations

I Size of limit cycle grows continuously from zero,and increases proportionally to

√µ− µc for µ close to µc.

I Frequency of limit cycle ≈ Im λ + O(µ− µc) near µc.

I Limit cycle is elliptical. Its shape becomes distorted as µ movesaway from µc.

I Eigenvalues follow curvy path (see figure).

� Subcritical Hopf Bifurcations in 2D

I Considerr = µr + r3 − r5, θ = ω + br2,

where cubic term is now destabilizing.

I µ < 0, 2 attractors with unstable limit cycle betweem them.

I As µ increases, unstable limit cycle tightens around origin,shrinks to 0 amplitude rendering it unstable at µ = 0.

I For µ > 0, Solutions near origin are forced to grow into large-amplitude oscillations.

� Example of Hopf Bifurcations in 2D

I Consider

x = µx− y + xy2, y = x + µy + y3.

I λ = µ± i. As µ increase through 0, origin changes from stable tounstable spiral. Hopf bifurcation.

I In polar coordinates, r ≥ µr + ry2. All trajectories are repelled to∞ for µ > 0. Can’t be supercritical.

I Can’t be degenerate: origin is not nonlinear center if µ = 0.

I Numerical integraton: subcritical.